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176. CMB 2003 (vol 46 pp. 98)

 Crossed Products by Semigroups of Endomorphisms and Groups of Partial Automorphisms We consider a class $(A, S, \alpha)$ of dynamical systems, where $S$ is an Ore semigroup and $\alpha$ is an action such that each $\alpha_s$ is injective and extendible ({\it i.e.} it extends to a non-unital endomorphism of the multiplier algebra), and has range an ideal of $A$. We show that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group $G$ of $S$ constructed in \cite[Proposition~6.1]{LR2}. It turns out that the full crossed product by this coaction is isomorphic to $A\rtimes_\alpha S$. If the coaction is moreover normal, then the isomorphism can be extended to include the reduced crossed product. We look then at invariant ideals and finally, at examples of systems where our results apply. Category:46L55

177. CMB 2002 (vol 45 pp. 321)

Brenken, Berndt
 $C^{\ast}$-Algebras of Infinite Graphs and Cuntz-Krieger Algebras The Cuntz-Krieger algebra $\mathcal{O}_B$ is defined for an arbitrary, possibly infinite and infinite valued, matrix $B$. A graph $C^{\ast}$-algebra $G^{\ast} (E)$ is introduced for an arbitrary directed graph $E$, and is shown to coincide with a previously defined graph algebra $C^{\ast} (E)$ if each source of $E$ emits only finitely many edges. Each graph algebra $G^{\ast} (E)$ is isomorphic to the Cuntz-Krieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix of~$E$. Categories:46LXX, 05C50

178. CMB 2002 (vol 45 pp. 265)

Nawrocki, Marek
 On the Smirnov Class Defined by the Maximal Function H.~O.~Kim has shown that contrary to the case of $H^p$-space, the Smirnov class $M$ defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, {\it i.e.} they have the same dual spaces and the same Fr\'echet envelopes. We describe a general form of a continuous linear functional on $M$ and multiplier from $M$ into $H^p$, $0 < p \leq \infty$. Keywords:Smirnov class, maximal radial function, multipliers, dual space, FrÃ©chet envelopeCategories:46E10, 30A78, 30A76

179. CMB 2002 (vol 45 pp. 309)

Xia, Jingbo
 Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ is compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$. Using local analysis in the full Toeplitz algebra $\calT = \calT (L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can be inferred from the compactness of a much smaller collection of commutators $[T_f,T_h]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper $C^\ast$-subalgebra $\calT (\calL)$ of $\calT$ which has the same essential commutant as that of $\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable of conferring the membership $f \in H^\infty + C(T)$ through the compactness of the commutators $\{[T_f,S] : S \in \calS\}$. Categories:46H10, 47B35, 47C05

180. CMB 2002 (vol 45 pp. 232)

Ji, Min; Shen, Zhongmin
 On Strongly Convex Indicatrices in Minkowski Geometry The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant---(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type. Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35

181. CMB 2002 (vol 45 pp. 3)

Azagra, D.; Dobrowolski, T.
 Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces. Categories:46B20, 58B99

182. CMB 2002 (vol 45 pp. 60)

Dranishnikov, A. N.; Gong, G.; Lafforgue, V.; Yu, G.
 Uniform Embeddings into Hilbert Space and a Question of Gromov Gromov introduced the concept of uniform embedding into Hilbert space and asked if every separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov's question negatively. Category:46C05

183. CMB 2002 (vol 45 pp. 46)

Dafni, Galia
 Local $\VMO$ and Weak Convergence in $\hone$ A local version of $\VMO$ is defined, and the local Hardy space $\hone$ is shown to be its dual. An application to weak-$*$ convergence in $\hone$ is proved. Categories:42B30, 46E99

184. CMB 2001 (vol 44 pp. 504)

Zhang, Yong
 Weak Amenability of a Class of Banach Algebras We show that, if a Banach algebra $\A$ is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of $\A$ implies the ($2m+1$)-weak amenability of $\A$ for all $m\geq 1$. Keywords:$n$-weak amenability, left ideals, left bounded approximate identityCategories:46H20, 46H10, 46H25

185. CMB 2001 (vol 44 pp. 355)

Weaver, Nik
 Hilbert Bimodules with Involution We examine Hilbert bimodules which possess a (generally unbounded) involution. Topics considered include a linking algebra representation, duality, locality, and the role of these bimodules in noncommutative differential geometry Categories:46L08, 46L57, 46L87

186. CMB 2001 (vol 44 pp. 370)

Weston, Anthony
 On Locating Isometric $\ell_{1}^{(n)}$ Motivated by a question of Per Enflo, we develop a hypercube criterion for locating linear isometric copies of $\lone$ in an arbitrary real normed space $X$. The said criterion involves finding $2^{n}$ points in $X$ that satisfy one metric equality. This contrasts nicely to the standard classical criterion wherein one seeks $n$ points that satisfy $2^{n-1}$ metric equalities. Keywords:normed spaces, hypercubesCategories:46B04, 05C10, 05B99

187. CMB 2001 (vol 44 pp. 335)

Stacey, P. J.
 Inductive Limit Toral Automorphisms of Irrational Rotation Algebras Irrational rotation $C^*$-algebras have an inductive limit decomposition in terms of matrix algebras over the space of continuous functions on the circle and this decomposition can be chosen to be invariant under the flip automorphism. It is shown that the flip is essentially the only toral automorphism with this property. Categories:46L40, 46L35

188. CMB 2001 (vol 44 pp. 105)

Pilipović, Stevan
 Convolution Equation in $\mathcal{S}^{\prime\ast}$---Propagation of Singularities The singular spectrum of $u$ in a convolution equation $\mu * u = f$, where $\mu$ and $f$ are tempered ultradistributions of Beurling or Roumieau type is estimated by $$SS u \subset (\mathbf{R}^n \times \Char \mu) \cup SS f.$$ The same is done for $SS_{*}u$. Categories:32A40, 46F15, 58G07

189. CMB 2000 (vol 43 pp. 418)

Gong, Guihua; Jiang, Xinhui; Su, Hongbing
 Obstructions to $\mathcal{Z}$-Stability for Unital Simple $C^*$-Algebras Let $\cZ$ be the unital simple nuclear infinite dimensional $C^*$-algebra which has the same Elliott invariant as $\bbC$, introduced in \cite{JS}. A $C^*$-algebra is called $\cZ$-stable if $A \cong A \otimes \cZ$. In this note we give some necessary conditions for a unital simple $C^*$-algebra to be $\cZ$-stable. Keywords:simple $C^*$-algebra, $\mathcal{Z}$-stability, weak (un)perforation in $K_0$ group, property $\Gamma$, finitenessCategory:46L05

190. CMB 2000 (vol 43 pp. 320)

Elliott, George; Fulman, Igor
 On Classification of Certain $C^\ast$-Algebras We consider \cst-algebras which are inductive limits of finite direct sums of copies of $C([0,1]) \otimes \Otwo$. For such algebras, the lattice of closed two-sided ideals is proved to be a complete invariant. Categories:46L05, 46L35

191. CMB 2000 (vol 43 pp. 368)

Litvak, A. E.
 Kahane-Khinchin's Inequality for Quasi-Norms We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about equivalence of $L_q$-norms of logconcave random variables (Kahane-Khinchin's inequality) to the quasi-convex case. We construct examples of quasi-convex bodies $K_n \subset \R$ which demonstrate that this equivalence fails for uniformly distributed vector on $K_n$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the tail" volume (for convex bodies such decay was proved by M.~Gromov and V.~Milman). Categories:46B09, 52A30, 60B11

192. CMB 2000 (vol 43 pp. 257)

Androulakis, George; Casazza, Peter G.; Kutzarova, Denka N.
 Some More Weak Hilbert Spaces We give new examples of weak Hilbert spaces. Category:46B45

193. CMB 2000 (vol 43 pp. 138)

Boyd, C.
 Exponential Laws for the Nachbin Ported Topology We show that for $U$ and $V$ balanced open subsets of (Qno) Fr\'echet spaces $E$ and $F$ that we have the topological identity $$\bigl( {\cal H}(U\times V), \tau_\omega \bigr) = \biggl( {\cal H} \Bigl( U; \bigl( {\cal H}(V), \tau_\omega \bigr) \Bigr), \tau_\omega \biggr).$$ Analogous results for the compact open topology have long been established. We also give an example to show that the (Qno) hypothesis on both $E$ and $F$ is necessary. Categories:46G20, 18D15, 46M05

194. CMB 2000 (vol 43 pp. 208)

Matoušková, Eva
 Extensions of Continuous and Lipschitz Functions We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant. As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$. Keywords:extension, continous, Lipschitz, Banach spaceCategories:54C20, 46B10

195. CMB 2000 (vol 43 pp. 193)

Magajna, Bojan
 C$^*$-Convexity and the Numerical Range If $A$ is a prime C$^*$-algebra, $a \in A$ and $\lambda$ is in the numerical range $W(a)$ of $a$, then for each $\varepsilon > 0$ there exists an element $h \in A$ such that $\norm{h} = 1$ and $\norm{h^* (a-\lambda)h} < \varepsilon$. If $\lambda$ is an extreme point of $W(a)$, the same conclusion holds without the assumption that $A$ is prime. Given any element $a$ in a von Neumann algebra (or in a general C$^*$-algebra) $A$, all normal elements in the weak* closure (the norm closure, respectively) of the C$^*$-convex hull of $a$ are characterized. Categories:47A12, 46L05, 46L10

196. CMB 2000 (vol 43 pp. 69)

Kaminker, Jerome; Perera, Vicumpriya
 Type II Spectral Flow and the Eta Invariant The relative eta invariant of Atiyah-Patodi-Singer will be shown to be expressible in terms of the notion of Type~I and Type~II spectral flow. Categories:19K56, 46L80

197. CMB 1999 (vol 42 pp. 274)

Dădărlat, Marius; Eilers, Søren
 The Bockstein Map is Necessary We construct two non-isomorphic nuclear, stably finite, real rank zero $C^\ast$-algebras $E$ and $E'$ for which there is an isomorphism of ordered groups $\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to \bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible with all the coefficient transformations. The $C^\ast$-algebras $E$ and $E'$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair of non-isomorphic, real rank zero, purely infinite $C^\ast$-algebras with similar properties. Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps, $C^\ast$-algebras, real rank zero, purely infinite, classificationCategories:46L35, 46L80, 19K14

198. CMB 1999 (vol 42 pp. 344)

Koldobsky, Alexander
 Positive Definite Distributions and Subspaces of $L_p$ With Applications to Stable Processes We define embedding of an $n$-dimensional normed space into $L_{-p}$, $0 Categories:42A82, 46B04, 46F12, 60E07 199. CMB 1999 (vol 42 pp. 321) Kikuchi, Masato  Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section~2 the equivalence between Shimogaki's theorem and some martingale inequalities will be established, and in Section~3 the equivalence between Boyd's theorem and martingale inequalities with change of probability measure will be established. Keywords:martingale inequalities, rearrangement invariant function spacesCategories:60G44, 60G46, 46E30 200. CMB 1999 (vol 42 pp. 221) Liu, Peide; Saksman, Eero; Tylli, Hans-Olav  Boundedness of the$q$-Mean-Square Operator on Vector-Valued Analytic Martingales We study boundedness properties of the$q$-mean-square operator$S^{(q)}$on$E$-valued analytic martingales, where$E$is a complex quasi-Banach space and$2 \leq q < \infty$. We establish that a.s. finiteness of$S^{(q)}$for every bounded$E$-valued analytic martingale implies strong$(p,p)$-type estimates for$S^{(q)}$and all$p\in (0,\infty)$. Our results yield new characterizations (in terms of analytic and stochastic properties of the function$S^{(q)}$) of the complex spaces$E$that admit an equivalent$q$-uniformly PL-convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the$L^p\$-boundedness of the usual square-function on scalar-valued analytic martingales. Categories:46B20, 60G46
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